A massive list of as much mathematical definitions and equations as possible in foundations of mathematics, pure mathematics, mathematical physics, applied mathematics, theoretical physics, applied physics, systems theory, information theory, engineering! **Foundations of Mathematics** * **Set Theory** * Zermelo-Fraenkel axioms (ZF) * Axiom of choice (AC) * Union, Intersection, Complement * Power Set * Cardinality * **Logic** * Propositional Logic, Predicate Logic * Truth Tables * Formal Proofs * Gödel's Incompleteness Theorems * **Number Theory** * Prime Numbers * Fundamental Theorem of Arithmetic * Modular Arithmetic * Fermat's Little Theorem * Euclidean Algorithm **Pure Mathematics** * **Calculus** * Limits * Derivatives * Integrals * Fundamental Theorem of Calculus * Differential Equations * **Linear Algebra** * Vectors, Matrices * Matrix Operations (addition, multiplication, etc.) * Determinants * Eigenvalues, Eigenvectors * Linear Transformations * **Abstract Algebra** * Groups, Rings, Fields * Isomorphisms, Homomorphisms * Galois Theory * **Topology** * Open Sets, Closed Sets * Metric Spaces * Homotopy * Fundamental Group **Mathematical Physics** * **Classical Mechanics** * Newton's Laws of Motion * Lagrangian Mechanics * Hamiltonian Mechanics * **Electromagnetism** * Maxwell's Equations * Lorentz Force Law * Coulomb's Law * **Quantum Mechanics** * Schrödinger Equation * Heisenberg Uncertainty Principle * Wave-Particle Duality * Dirac Equation * **Special & General Relativity** * Lorentz Transformations * Einstein's Field Equations * Spacetime Curvature **Applied Mathematics** * **Probability** * Probability Distributions (Normal, Poisson, etc.) * Bayes' Theorem * Central Limit Theorem * **Statistics** * Hypothesis Testing * Confidence Intervals * Regression Analysis * **Numerical Analysis** * Root-finding algorithms * Integration Methods (Quadrature) * Interpolation Techniques **Theoretical Physics** * **Quantum Field Theory** * Path Integrals * Feynman Diagrams * Renormalization * **String Theory** * Bosonic String theory * Superstring Theory * M-Theory **Applied Physics** * **Solid State Physics** * Band Structure * Semiconductors * Superconductivity * **Optics** * Laws of Reflection and Refraction * Lens Equations * Diffraction and Interference * **Thermodynamics** * Laws of Thermodynamics * Entropy * Heat Engines **Systems Theory** * **Dynamical Systems** * Attractors, Bifurcations * Chaos Theory * **Control Theory** * State-space Representation * Feedback Control * Stability Analysis **Information Theory** * **Entropy (Information Theory)** * **Shannon's Channel Coding Theorem** * **Data compression** **Engineering** * **Electrical Engineering** * Kirchhoff's Laws * Ohm's Law * AC Circuit Analysis * **Mechanical Engineering** * Stress and strain * Thermodynamics cycles * Fluid dynamics (Bernoulli's Equations) ### Foundations of Mathematics - Set: A collection of distinct objects. - Function: A relation between a set of inputs and a set of permissible outputs. - Cardinality: The number of elements in a set. - Axiom of Choice: Every non-empty product of non-empty sets is non-empty. - Zermelo-Fraenkel set theory (ZF) - Axiom of Extensionality: Two sets are equal if they have the same elements. - Peano axioms for natural numbers - Gödel's incompleteness theorems - Cantor's theorem: \(|2^A| > |A|\) for any set \(A\). ### Pure Mathematics #### Algebra - Group: A set equipped with an operation that combines any two of its elements to form a third element. - Ring: A set equipped with two binary operations satisfying properties analogous to addition and multiplication. - Field: A ring in which division is possible, excluding division by zero. - Vector space: A collection of vectors, which may be added together and multiplied by numbers. #### Geometry - Euclidean geometry: Based on the postulates of Euclid. - Manifold: A topological space that locally resembles Euclidean space. - Riemannian geometry: Studies smooth manifolds with a Riemannian metric. #### Analysis - Limit: The value that a function or sequence "approaches" as the input or index approaches some value. - Continuity: A function is continuous if, roughly speaking, small changes in the input result in small changes in the output. - Derivative: The rate at which a function is changing at any given point. - Integral: A generalization of area under a curve. #### Number Theory - Prime number: A natural number greater than 1 that has no positive divisors other than 1 and itself. - Fermat's Last Theorem: \(x^n + y^n = z^n\) has no non-zero integer solutions for \(x\), \(y\), \(z\), and \(n > 2\). - Riemann Hypothesis: All non-trivial zeros of the Riemann zeta function have real part 1/2. ### Mathematical Physics - Schrödinger equation: \(i\hbar\frac{\partial}{\partial t}\Psi(x,t) = H\Psi(x,t)\) - Einstein's field equations: \(G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}\) - Maxwell's equations: Describe the fundamentals of electricity and magnetism. ### Applied Mathematics - Fourier transform: Transforms a function of time into a function of frequency. - Navier-Stokes equations: Describe the motion of viscous fluid substances. - Black-Scholes equation: Used to model the dynamics of financial markets. ### Theoretical Physics - General relativity: \(G_{\mu\nu} = 8\pi T_{\mu\nu}\) - Quantum mechanics: Wave-function, \(|\Psi|^2\) represents the probability density. - Standard Model of particle physics: Theory describing three of the four known fundamental forces. ### Applied Physics - Ohm's law: \(V = IR\) - Hooke's law: \(F = -kx\) - Thermodynamics: First law (\(\Delta U = Q - W\)), Second law (entropy). ### Systems Theory - Feedback loop: A system structure that causes output from one node to eventually influence input to that node. - Control theory: Mathematical study of the control of dynamical systems. - Cybernetics: The study of regulatory systems. ### Information Theory - Entropy: A measure of the uncertainty in a set of outcomes. - Shannon's Theorem: Establishes fundamental limits on signal processing operations, such as compressing data and reliably transmitting data over noisy channels. - Mutual Information: A measure of the amount of information that two variables share. ### Engineering - Newton's laws of motion: Fundamental principles describing the relationship between the motion of an object and the forces acting on it. - Bernoulli's principle: An increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. - Kirchhoff's circuit laws: Laws that deal with the conservation of charge and energy in electrical circuits. **Foundations of Mathematics** * **Category Theory** * Objects, Morphisms * Functors, Natural Transformations [Image of a basic category theory diagram] * **Type Theory** * Lambda Calculus * Dependent Types * Homotopy Type Theory **Pure Mathematics** * **Real Analysis** * Sequences and Series * Bolzano-Weierstrass Theorem * Uniform Continuity * **Complex Analysis** * Cauchy-Riemann Equations * Analytic Functions * Riemann Surfaces [Image of a Riemann Surface] * **Differential Geometry** * Manifolds * Riemannian Metrics * Curvature [Image of a manifold] * **Algebraic Geometry** * Algebraic Varieties * Schemes * Sheaf Theory **Mathematical Physics** * **Statistical Mechanics** * Boltzmann Distribution * Partition Function * Phase Transitions * **Fluid Dynamics** * Navier-Stokes Equations [Image of Navier-Stokes Equations] * Vorticity * Turbulence **Applied Mathematics** * **Optimization** * Linear Programming * Convex Optimization * Gradient Descent * **Graph Theory** * Shortest Path Algorithms (Dijkstra, Bellman-Ford) * Graph Coloring * Network Flows [Image of a graph in Graph Theory] * **Cryptography** * RSA Encryption * Elliptic Curve Cryptography **Theoretical Physics** * **Cosmology** * Friedmann Equations * Big Bang Model * Cosmic Inflation * **Particle Physics** * Standard Model of Particle Physics * Higgs Mechanism * Quantum Chromodynamics (QCD) **Systems Theory** * **Game Theory** * Nash Equilibrium * Prisoner's Dilemma * Auction Theory [Image of Prisoner's Dilemma payoff matrix] **Information Theory** * **Signal Processing** * Fourier Transforms [Image of Fourier Transform] * Wavelets * Filter Design * **Coding Theory** * Error-Correcting Codes * Hamming Codes **Engineering** * **Structural Engineering** * Beam Theory * Finite Element Analysis (FEA) * Earthquake Engineering * **Aerospace Engineering** * Rocket Equation * Orbital Mechanics * Aerodynamics ### Foundations of Mathematics (Continued) - Russell's Paradox: Shows that some sets cannot be members of themselves. - Banach-Tarski Paradox: A ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. - Turing Machine: A mathematical model of computation that defines an abstract machine. ### Pure Mathematics (Continued) #### Topology - Homotopy: A continuous transformation of one function or shape into another. - Fundamental group: Describes the basic loop-like structures in a space. - Compactness: A property that generalizes the notion of a subset of Euclidean space being closed and bounded. #### Combinatorics - Graph Theory: The study of graphs, which are mathematical structures used to model pairwise relations between objects. - Pigeonhole Principle: If \(n\) items are put into \(m\) containers, with \(n > m\), then at least one container must contain more than one item. - Ramsey Theory: Studies conditions under which order must appear. #### Logic - Propositional logic: Deals with propositions and their connectives. - Predicate logic: Expands on propositional logic by dealing with predicates and quantifiers. - Model theory: Studies the models of various theories. ### Mathematical Physics (Continued) - Hamiltonian mechanics: A formulation of classical mechanics. - Lagrangian mechanics: Another formulation of classical mechanics, equivalent to Newton's laws. - Quantum field theory: The theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics. ### Applied Mathematics (Continued) - PDE (Partial Differential Equations): Equations that involve rates of change with respect to continuous variables. - Optimization: The selection of the best element from some set of available alternatives. - Game theory: The study of mathematical models of strategic interaction among rational decision-makers. ### Theoretical Physics (Continued) - String theory: A theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects known as strings. - Loop Quantum Gravity: A theory that attempts to describe the quantum properties of the universe and gravity. - Supersymmetry: A proposed type of space-time symmetry that relates two basic classes of elementary particles: bosons and fermions. ### Applied Physics (Continued) - Semiconductor equations: Describe the electrical behavior of semiconductors, including the diode and transistor. - Laser theory: Describes the operation of lasers, based on the principles of quantum mechanics. - Photovoltaic effect: The generation of voltage and electric current in a material upon exposure to light. ### Systems Theory (Continued) - Dynamical systems: Mathematical models used to describe the time-dependent position of a point in a given space. - Chaos theory: The study of dynamical systems that are highly sensitive to initial conditions. - Network theory: The study of graphs as a representation of either symmetric relations or asymmetric relations between discrete objects. ### Information Theory (Continued) - Coding theory: Deals with the design of error-correcting codes for the reliable transmission of information across noisy channels. - Channel capacity: The tightest upper bound on the amount of information that can be reliably transmitted over a communication channel. - Source coding theorem: Establishes the limits of possible data compression. ### Engineering (Continued) - Fluid dynamics: The study of fluids (liquids and gases) in motion. - Material science: The study of the properties of solid materials and how they can be used in various products. - Structural analysis: The determination of the effects of loads on physical structures and their components.